# metric

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## metric

**1.**of or relating to the metre or metric system

**2.**

*Maths*denoting or relating to a set containing pairs of points for each of which a non-negative real number ρ(

*x, y*) (the distance) can be defined, satisfying specific conditions

**3.**

*Maths*the function ρ(

*x, y*) satisfying the conditions of membership of such a set (a

**metric space**)

## Metric

a mathematical term that denotes the rule for determining a given distance between any two points (elements) of a given set *A*. A real number function that satisfies the following three conditions is called the distance ρ (*a, b*) between the points *a* and *b* of the set *A:* (1) ρ (*a, b*) ≧ 0, where ρ(*a, b*) = 0 if and only if *a* = *b*; (2) ρ(*a, b*) = ρ(*b, a*); and (3) ρ(*a, b*) + ρ(*b, c*) ≧ ρ(*a, c*). For a given set *M*, a metric may be introduced in other ways. For example, on a plane we may take not only the ordinary Euclidean distance

as the distance between points *a* and *b* having coordinates (*x*_{1}, *y*_{1}) and (*x*_{2}, *y*_{2}), respectively, but we may take other distances as well, for example,

ρ_{1}(*a, b*) = ǀ*x _{1}* –

*x*

_{2}ǀ + ǀ

*y*

_{1}–

*y*

_{2}ǀ

In (functional and coordinate) vector spaces a metric is often defined as a norm or, sometimes, as a scalar product. In differential geometry, a metric is introduced by specifying an element of arc length by means of a differential quadratic form. A set that has a metric introduced in it is referred to as a metric space.

A metric is sometimes understood to denote a rule for determining not only distances but also angles; an example is a projective metric.

V. I. SOBOLEV

## metric

[′me·trik]*X*satisfying four rules: for

*x, y,*and

*z*in

*X,*the distance from

*x*to itself is zero; the distance from

*x*to

*y*is positive if

*x*and

*y*are different; the distance from

*x*to

*y*is the same as the distance from

*y*to

*x*; and the distance from

*x*to

*y*is less than or equal to the distance from

*x*to

*z*plus the distance from

*z*to

*y*(triangle inequality).